The limitations of the Poincar{é} inequality

Abstract

We examine the validity of the Poincaré inequality for degenerate, second-order, elliptic operators $H$ in divergence form on $L_2(\Ri^{n}\times\Ri^{m})$. We assume the coefficients are real symmetric and $a_1{H}_{\hat{I}}´\ge H\ge a_2{H}_{\hat{I}}´$ for some $a_1,a_2>0$ where $H_{\hat{I}}´$ is a generalized GruÅ¡in operator, \[ H_δ=-\nabla_{x_1}\,|x_1|^{(2δ_1,2δ_1')}\,\nabla_{x_1}-|x_1|^{(2δ_2,2δ_2')}\,\nabla_{x_2}^2 \;. \] Here $x_1\in\Ri^n$, $x_2\in\Ri^m$, $\hat{I}{´}_1,\hat{I}{´}_1^{\prime }\in [0,1⟩$, $\hat{I}{´}_2,\hat{I}{´}_2^{\prime }\ge 0$ and $|x_1{|}^{(2\hat{I}´,2\hat{I}{´}^{\prime })}=|x_1{|}^{2\hat{I}´}$ if $|x_1|\le 1$ and $|x_1{|}^{(2\hat{I}´,2\hat{I}{´}^{\prime })}=|x_1{|}^{2\hat{I}{´}^{\prime }}$ if $|x_1|\ge 1$. \smallskip We prove that the Poincaré inequality, formulated in terms of the Riemannian geometry corresponding to $H$, is valid if $n\ge 2$, or if $n=1$ and $\hat{I}{´}_1\vee \hat{I}{´}_1^{\prime }\in [0,1/2⟩$ but it fails if $n=1$ and $\hat{I}{´}_1\vee \hat{I}{´}_1^{\prime }\in [1/2,1⟩$. The failure is caused by the leading term. If $\hat{I}{´}_1\in [1/2,1⟩$ it is an effect of the local degeneracy $|x_1{|}^{2\hat{I}{´}_1}$ but if $\hat{I}{´}_1\in [0,1/2⟩$ and $\hat{I}{´}_1^{\prime }\in [1/2,1⟩$ it is an effect of the growth at infinity of $|x_1{|}^{2\hat{I}{´}_1^{\prime }}$. If $n=1$ and $\hat{I}{´}_1\in [1/2,1⟩$ then the semigroup $S$ generated by the Friedrichs' extension of $H$ is not ergodic. The subspaces $x_1\ge 0$ and $x_1\le 0$ are $S$-invariant and the Poincaré inequality is valid on each of these subspaces. If, however, $n=1$, $\hat{I}{´}_1\in [0,1/2⟩$ and $\hat{I}{´}_1^{\prime }\in [1/2,1⟩$ then the semigroup $S$ is ergodic but the Poincaré inequality is only valid locally. \smallskip Finally we discuss the implication of these results for the kernel of the semigroup $S$.